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This is a game for two players. You will need some small-square
grid paper, a die and two felt-tip pens or highlighters. Players
take turns to roll the die, then move that number of squares in a
straight line. Move only vertically (up/down) or horizontally
(across), never diagonally. You can cross over the other player's
trails. You can trace over the top of the other player's trails.
You can cross over a single trail of your own, but can never cross
a pair of your trails (side-by-side) or trace over your own trail.
To win, you must roll the exact number needed to finish in the
target square. You can never pass through the target square. The
game ends when a player ends his/her trail in the target square, OR
when a player cannot move without breaking any of the rules.

Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?

What Does Random Look Like?

Stage: 3 Challenge Level:

Why do this problem?

This problem is one of a set of problems about probability and
uncertainty. Intuition can often let us down when working on
probability; these problems have been designed to provoke
discussions that challenge commonly-held misconceptions. Read more
in this
article.

This problem requires learners to make sense of experimental data
and graphical representations.
The probabilities associated with coin flipping allow learners to
analyse and explain the distributions that emerge, and get a feel
for the features they would expect a random sequence to
exhibit.

Possible approach

Hand out two of these
strips to each learner. Ask everyone to make up a sequence of
Hs and Ts as if they came from a sequence of coin flips, and to
write it down on their first strip, writing "made up" lightly in
pencil on the back of the strip. Then ask everyone to flip a coin
twenty times and record each outcome on the second strip, writing
"real" on the back.

Arrange the learners in groups of three or four, and ask each
group to swap ALL their strips with another group, and then
challenge them to sort the strips into two piles, "real" and "made
up", WITHOUT looking at the back of the strips.

Once every group has had a chance to do this, they can turn
over the strips to see how many they got right. Take some time to
discuss any criteria they used to decide.

If you have
access to computers:

Ask learners to work in pairs at
the computer and give them time to explore the animation. Then ask
them to generate several sequences of twenty coin flips and try to
get a feel for the features they would expect a random sequence to
have. If necessary, suggest that they consider averaging the number
of runs of length 2, 3, 4 and so on.

If you don't
have access to computers:

If possible, show the interactivity
to the whole class and ask them to try to make sense of the bar
chart. If this isn't possible, write up a random sequence generated
with a coin, and demonstrate the way that the interactivity counts
runs.

Ask each group to plot graphs in
the same way for the randomly generated sequences they created at
the start of the lesson. Once they have done this, encourage them
to compare their graphs and identify the key features, perhaps
suggesting that they find the average number of runs of length 2,
3, 4 and so on within their group.

Bring the class together and discuss the key features of the random
sequences that they found, as well as any explanations of why the
run lengths were distributed the way they were, referring to the
probabilities of $\frac{1}{2}$ and $\frac{1}{4}$ and so on
associated with coin flipping.

Finally, ask each group to give their original real and made-up
strips to a DIFFERENT group from the one they swapped with before.
Can they use their new-found insights to spot the fakes
successfully?

Key questions

What proportion of the time would you expect to flip the same
as you got on the previous flip?

What proportion of the time would you expect to flip the same
as you got on the TWO previous flips?

Possible extension

The problem Can't
Find a Coin challenges learners to fool the computer with a
sequence of 100 coin flips.

Possible support

Give learners lots of time to explore the interactivity and make
sense of the different bars on the bar chart.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.